Nature & Number, Pythagoras & Fudge

A vibrating string, a perfect structure...

In his Lives of the Artists, Giorgio Vasari writes of Filippo Brunelleschi, the architect of the great dome on the cathedral in Florence,

…we should never turn up our noses when we meet people who in their physical appearance do not possess the initial grace and beauty that Nature should bestow upon skillful artisans when they come into the world…

Filippo was a great genius, but not all that good looking. Note the use of the word should in the phrase “…that Nature should bestow upon skillful artisans.” How can Nature do anything that it should not do? Just whose rules does Nature obey, if not its own? The idea here, that Nature has done something wrong, made a mistake, had a little hiccough, in making a great genius an ugly man, or at least, insignificant in appearance, may be common to Italy, or to Renaissance thinkers, but it is also part of an immensely deep and broad current of thought in western culture since the Ancients. Outward beauty reflects inward perfection. Personal beauty is a manifestation of the soul’s purity.

“Beauty is truth, truth beauty,” – that is all
Ye know on earth, and all ye need to know.

And if beauty is the emanation of the soul, why should an artistic genius not be beautiful? How could it happen? It’s a violation of the nature of the universe, the ordered universe in which truths are manifest in the order and lovliness of things. And the most beauteous things of all are the pure things, the mathematical entities, the Pure Forms of Plato, the Ideas.

Historians of ideas (Man of Roma included) agree that this great torrent of intellectual traditions has its source with Pythagoras, the student of Thales, and a predecessor of Plato and Socrates. He was a brilliant thinker, a mystic, an analyst, a mathematician, and the founder of a cult that has lived on to our day in various forms. In the wonderful collection of brief mathematical lives, Gems of Calculus, he is referred to as 3/5 genius, 2/5 sheer fudge . Bertrand Russell appraises him similarly, and that is no small compliment!

From his followers’ mystical preoccupation with Number, and awestruck encounters with the order of the universe, Plato developed his metaphysical notions, Platonists mixed Plato with eastern cults, early Christians mixed Plato with Christ, later Christians mixed it all up into neo-Platonism, the Renaissance rediscovered paganism, Platonism, and mysticism allied to the beauty of art, and secular and mystical philosophers of the succeeding ages remained in thrall to the notions of:

A universe explicable in terms of number (Do I have to point out the obvious pop-Pythagorean nightmare – The Matrix, and Keanu/Neo?)

Beauty founded on elegance of formulation, mathematical economy, and aximomatic inevitability

The nature of the universe revealed through intellectual intuition and analysis and NOT through experience

The truth as imminent, but not obvious

Just a listing of these notions evokes so many associations, it’s clear Mr. P. was onto something big. Did he invent these ideas? Probably not. But he was the first to articulate them in a way that had sticking power in the Western tradition. I would guess that these notions have their roots far deeper, in the human organisms evolution as an information processing being. The intellectual excitement of these ideas is a refined form of the fundamental “Aha!” feeling that comes with discovery…of food…of the lever…of the power of fire….

In our own day, these ideas live on, certainly in religious rhetoric, but they are also increasingly problematic as I shall discuss later. Consider just Thomas Pynchon’s novel, V,in which a young woman, Esther, is having an affair with a plastic surgeon, Dr. Shale Shoenmaker (Dr. Shale Beauty-maker). He wants to give Esther a nose job, she is not sure why she should have one. It’s a popular comic theme from the 1950’s. The Mad Magazine song goes,

“I once knew a girl with such a big schnoz,
she couldn’t get a boyfriend, or a job!
So she got a nose job!
Yeah, yeah, yeah!

The good doctor tells Esther that he wants to bring out the true beauty within her, make her outward experience in harmony with the inward nature of her soul, rectify, improve on the work of nature. It is a pure Renaissance Neo-Platonic argument about art, truth, and beauty, but he was being ruthlessly satirical. And of course, in our Botoxed present, who can deny that we have gotten something a bit wrong with the beauty-truth-body equation?

How did this concept get going? It was the music. The Pythagoreans noticed that by causing a string to vibrate and sound a tone, they could create pleasing scales of tones by holding down the string at set increments of its length, effectively shortening the string. Thus, the musical intervals were codified, if not quite born. Simple ratios, pleasing scales. In the visual realm, pleasing proportions, the Golden Mean, which lives on in dimensions of our rooms and the size of a standard piece of writing paper.

Something always puzzled me about this, however, since I am not musical. How did they know the scales were pleasing, were right? They just heard it, but other people heard differently. Asian tonal scales are not the same as ours. The de-tuned scales of the blues and other genres are pleasing to their audiences, but hardly classical. My nom de plume, Lichanos (more in my By Way of Explanation), refers to a particular ancient scale. Was it a deviant one? Was there some fudging of the scales at the creation? Did they weed out the not-quite-right tones so that only the ones with “good” ratios remained? I await the response of the archaeo-musicologists amongst you!

I said earlier that this current of thought is not always good – it was very much an impediment to the development of science. Ideas that were not “beautiful” were discarded. Ideas not deduced from geometry and pure forms were considered suspect. Even in Newton’s day, he felt he must prove his theories twice: once as geometrical demonstrations that fill the pages of his Principia, and once in terms of argument that are derived from his laws of motion and his observations. In science, the truth is not always the beauty of Pythagoras and Plato. Avogadro’s Number, without which we cannot solve chemical equations, is an ugly number. Planck’s constant is not pretty either. The acceleration of gravity (32.2 feet per second per second) is not lovely. Don’t even mention the contant of universal gravitational attraction! Even so, the lure of number remains, as an obstacle and as a motivation for science.

This split between two modes of apprehending the universe is represented in Raphael’s famous image of the School of Athens. Plato gestures upward towards the empyrean realm. Aristotle points downward towards the earth.

Plato dominated western thought until the great resurrection of The Philosopher, as the schoolmen called Aristotle, in the 12th century Renaissance. He had his own scientific “issues” and the reign of Platonism was by then, in any case, well established.

The seduction of the geometric! The fact that geometry revealed incommensurable, irrational numbers only placed a slight speed bump in front of the onward rush of the Pythagorean fleet.

If a=1, and b=1, then c=square root of 2. Punch that into your calculator and see what a nice, beautiful number you get! Still, the Numbers as the final reach of truth, the ultimate ground, the thing in itself carried on. How was it that mathematics could tell us about this earthly realm? Fire a missile, from a canon or a slingshot, and it follows a parabolic arc (Gravity’s Rainbow) to its final resting place. We can predict its path precisely – why? The disjunction between experience and the pure realm of mathematics is bridged in physics, but how? Kant wrestled with this and concluded…well, another time.

Today, the debased form of this issue lives on, melded with religious fundamentalism, in the argument against Darwin from Intelligent Design. How could the world be anything but designed, according to plan? There is a whiff of the old pagan, Pythagoras, here.

For this, for everything, we are out of tune;
It moves us not. — Great God! I’d rather be
A Pagan suckled in a creed outworn;
So might I, standing on this pleasant lea,
Have glimpses that would make me less forlorn;
Have sight of Proteus rising from the sea;
Or hear old Triton blow his wreathed horn.

It’s the last part of the poem only. I think he’s saying that since we live in a materialistic and unspiritual age, if he were a pagan, a truly believing one, although he “knows” that creed is “outworn,” his glimpses of nature’s wonders would not sadden him as they do now. He would feel “in tune” with them – there go those strings hummin’ again! – unlike we modern folks.

It is a real pleasure to be in conversation here. I hope my ignorance doesn’t grate too much.

May I have your views as to the extent you regard the number system, from natural numbers through irrational numbers, real numbers, transcendental numbers, imaginary numbers to transfinite numbers is a human invention, and if so, why they accord so much with the behaviour of the natural world.

Will Rogers, great American comic said, “We are all ignorant, just of different things.” I try to live by that.

The question you ask is so big, so fundamental, and also not one I have devoted much thought too simply because it doesn’t intrigue me too much. Others feel differently!! I do have thoughts on why mathematics accords with the natural world, nothing much on number theory. I’ll try and post about them later if I can sustain the energy to work through my ideas. It’s been awhile that I’ve been on that one…

After Phytagoras was long dead, the Athenian aristocrats saw their educational ideal (paideia) in the phrase καλὸς καὶ ἀγαθός, kalos kai agathos, beautiful and good, that made the superior being (not in the Nazi sense tho, THAT was an invention from our sweet Spartans).

This ideal is possibly linked to P’s legacy (tho not only, a deep sense of beauty was in most of the Greeks: their towns scattered all over the Med were always both commercially and beautifully located, while to the Phoenicians only the commercial factor mattered.

Beautiful and Good we were saying. The sense was a bit different from ours. It added (Socrates’ clarification) Truth, to Good and Beauty, which became a trinity-unity since to Socrates Good was Truth (knowledge, wisdom) while Evil was ignorance. I know you meant that too, I just wanted to point it out ah ah ah.

PS
Of course truth was into Pythagoras and his math-geometry. The problem with Socrates is we know him only through Plato. (Same with Pyth, he didn’t like to write so we know him via his pupils who were so many)

I would guess that these notions have their roots far deeper, in the human organisms evolution

I think this to be very interesting. Also argumentation or reason (logos) is are possibly inborn but the merit of the Greeks was to make those skills explicit with attached a rich toolbox to make arguments more solid etc. (but we have Chinese and Indian arguments too…)

Something always puzzled me about this, however, since I am not musical. How did they know the scales were pleasing, were right? They just heard it, but other people heard differently. Asian tonal scales are not the same as ours.

It is not a matter of scales but of harmonics, which pertain to physics, not to doxa, opinion or subjectivity. Naturally, with scales, every culture has its own scales, a scale is a fixed sequence of tones or pitches on which melodies are built upon and that is felt beautiful only to those that are used to it.

Only consider that P made experiments on the string of Greek lyrer. A string has a big advantages over, say, a flute, since harmonic sounds come out more easily much depending on the way you touch such string).

Phythagoras and his pupils were no fools. They were good (ethics), possibly beautiful or in any case knew how to make a body healthy and better (aesthetics), and they seeked truth (scientific? Heuristic? Epistyemology is not my forte). They could not build but on rock all their mystical scientific religion.

Purification via science. I think also Socrates was Phythagorean after all.

Yes, harmonies are not subjective, but a physical phenomemon. I realized this – see my comment to Troutsky above where I corrected.

I am not at all clear on the notions of scale, tone, harmony, octave, step, etc. etc. As I said, I am a musical illiterate, a major weakness in my education which I am too lazy to remedy, despite the occasional help of my wife.

I should get a Music Theory for Dummies or something!

But…if P and friends were amazed at the relationship of harmony (sensed by the ear) and string-length ratios, that, to me, is not very amazing. The ear can sense harmony – it too is a vibrating thing. I guess what really confuses me is scales, half-steps, full steps, tuning, etc. etc.

I partially agree with the statement that outer beauty reflects inner purity….i think we should again consider what is meaning of beauty…..it may differ person by person…to connect purity with beauty i personally think is not possible..look at a small grass flower..it may not be beautiful,but it is pure…..or consider the famous mystic,ashtawakra.his body was bent at 8 places.he was worst creature to have a look at.but i doubt that there is no one more pure than him.

@philomystics
Well, as far as I am concerned, when I said ‘I agree’ I meant ‘I agree that for Phythagoras and many other Greeks that was important’ and that it was a legacy from him and from them brought to the West. Just trying to reconstruct here the history of conceptions – I mean – more than criticizing conceptions themselves.

History is important because the way we were impacts on the way we are. For example positive heroes /heroines even in fables have always been beautiful ‘and’ good while the evil ones were ugly ‘and’ bad – take Cinderella: that is Greek legacy too. This was from classical antiquity (Homer etc.) until 1800 CE, with Romanticism, with Gothic novels etc. From that period on in fact the ugly could become not just evil but something else. Victor Hugo is a good example.

I must point out that it was Romanticism that “rescued” the ugly from spiritual desolation. Quasimodo was, in fact, pure inside, despite his hideous appearance. Is that what you meant with your reference to Hugo?

@LichanosQuasimodo was, in fact, pure inside, despite his hideous appearance. Is that what you meant with your reference to Hugo?

Yes. I meant Quasimodo.

Umberto Eco btw has dedicated some time to this topic, as his ‘On Beauty: A History of a Western Idea’ attests. Also his last essay (one month ago? Forgot the title) is dedicated to this topic. I read neither of them, but leafed reviews on both. Eco is a medievalist, ex Catholic, scholar of Joyce not by chance, and he likes aesthetics quite a lot. He was absolutely my man for a while, when I was 35-40, but after ‘Il nome della rosa’ – which I loved – I got tired of him. His theoretical books (plus some lovely articles on America, California most of all: ‘Dalla periferia dell’Impero’) were exciting and revealing at that time. Now I find him boring. But I might get back to some of his works now that the Middle Ages more appeal (a little bit, not that much) for my dilettante research.

I’d explain for possible readers – not certainly for you – that the reason Plato’s gesture is upwards is that, to him, knowledge came from preconceived ideas (the base of Plato’s thought). Such ideas, as you said, reside in a world ‘beyond’ physics (ie meta-physics).

Aristotle instead points in front of him, meaning that especially via experience (the real world, not a world beyond) we get to form our ideas generalizations & categories through which we acquire knowledge, then again from ideas to experience and from it back to ideas etc.

And I’d add that (this time for readers and for you both lol) – since you all out there ‘culturally’ come from the Anglo Saxons no matter what (we’ve discussed in my blog how genes matter not that much :-) – in the history of thought the British people embraced Aristotle more than the folks in the ‘Continent’, ie Italy France Spain, Germany, Austria and the rest.

Every folk borrows what is more congenial to its mind: the Brits were more matter of fact (in the best sense of the term), like the Romans were. Aristotle better appealed to the British mind. Possibly – but I an ass in modern philosophy – empiricism, US pragmatism etc. vs German Italian French rationalism, this difference I mean came from the fundamental bifurcation of (Socrate’s?) thought: Plato, S’s most creative heir, and Aristotle, Plato’s most creative heir.

Of course Pythagoras enters the equation quite a lot, although there sources a very contradictory. We know a lot but little about him (like Socrates)

After Cicero’s Somnium Scipionis – a wonderful beautiful comsology tale – I am now reading Plato’s Timaeus – an even more wonderful tale – one of Plato’s last dialogues.

Well, is it by chance that Timaeus was an Italian Greek from Locri (a Calabrian polis where Pythagoreanism had deep roots) and that this guy explained in a dense monologue Pato’s view of the universe in its most complete, poetic, inspired, musical, harmonious way?

PS
Both rationalism (Descartes, Leibniz, Spinoza, Kant ) and Anglo Saxon empiricism, pragmatism etc. have pros and cons. A friend of mine went (40 years a go) to Live and work in Cambridge university (UK). He lived there for 8 years. He kept complaining in his letters to me: “These people are too matter of fact. Their philosophy is so disappointing”. I know he was tired of living there for so long. But I understand what he meant.

Yes, I’ve posted a lot about the pitfalls of empiricism. When I was in university, and so disgusted with what I saw as their narrow, dogmatic view of things, (those practitioners of analytic philosophy, as they called it…) I coined the phrase, Running Dog Lackeys of the Empiricist Scourge. But for years, I’ve considered myself a follower of David Hume, who avoided their pitfalls, mostly because of his temperament, I think: See Hume1 & Hume2

On the other hand, the Continental philosophers tend to take their marching orders from Plato-Kant-Hegel-Marx-Freud, and have created their own bizarre dogma which reaches its apogee in today’s “deconstructionist” approach that has contempt for science and experience. In all, I think building on Hume and science is the better way, while not being too much of a “beef eater.” (The English have been derided as shopkeepers and blockhead-beef eaters by some Continentals. See this print by my hero, James Gillray for a wonderful send up this idea that cuts both ways! French Liberty – British Slavery)

Have you read Karl Popper (The Open Society and Its Enemies?) A classic, a monumental demolition of the influence of Plato (vol. I) and Hegel (vol. II) on Western thought. I think he tried to follow the best tradition of Hume.

I read some Popper long time ago. My wife is into epistemology. I am not qualified enough.

Let me be the Continent’s advocate just for the hell of it. Mind I’m not basely trying captatio belevolentiae (the Jewish Continental factor), it being a fact.

The continental rationalist Plato-Kant-Hegel-Marx-Freud chain (Descartes too possibly) has produced totalitarian minds. The Anglo Saxon (UK and US) approach has produced more liberty for sure but also – it is my view only- a tiny bit of flatness in thought.

As for concrete scientific discoveries both the Continent and the Anglo Saxons have produced a lot. True, America eats up all noble prizes, but many scientists were from Europe (the Continent lol? Today from Asia etc. bla bla bla)

UK produced huge towers such as Darwin and others, even ad advocate must retain some decency ;-)

I observe it took Popper, a Viennese ‘continental’ Jew to come to terms with a lot of philosophical problems the Anglo Saxons could not solve.

I observe the Anglo-Saxons (the US mainly) would have pained A LOT (10 years? 15 years?) to do thorough Jap clean-up in Asia weren’t for a long series of ‘continental again’ Jews – who made the A-bomb a) conceivable and 2) helped actually ‘making’ it in due time – starting from Einstein and ending up with Enrico Fermi, a genuine Roman Jew – allow me ah ah ah – whose Via Panisperna school of physics is just a few yards from my home – we have discussed the ancient-ness of these people from both a Jewish & Roman angle (ad nauseam.)

[Btw, the non Continental Jews, were they sleeping by any chance? ;-) Isn’t insularity an overall problem with angli and their – cultural – descendants? I mean, take the UK, NZ, Australia, US … hope you won’t get offended .. a huge world tho made of islands, isn’t that damn interesting? A theme worth some discussion some day in my view. I mentioned that to a Canadian blogger, he got upset about it …he is such a good boy tho – not Paul in any case ]

As for Popper again, I’ll add I like his idea of open society, but I think his criticism of historicism to be … too logical … thence abstract. Historicism has many flaws, without a doubt – but unfortunately the roots – my own – I care for more are right there, in this genuinely Italian (and a tad French too) parochial line made of Croce-Gramsci-Magister-French Annales bla bla.

Croce, a Neapolitan version of Kultur. Gramsci – foundational to me also because of M, a sort of neo G – a Sardinian (who studied and lived in Piedmont tho) son of Machiavelli, Marx, Labriola, Sorel, Pareto, Bergson, Croce and Gentile.

And, this science non-science thing … I understand it to be important, provided we do not have the religion of science, such a poor thing, science as a religion, allow me.

Science is … just a tool [like mysticism?] It produces exceptional results, but it is not a ‘good for all’ tool, no tool, experience can be. Maybe – as bizarre Extropian likes to say – in 5000 thousand years …. but I will be dead I presume.

A. WTH is blockhead-beef eaters?
B. Many French intellectuals since 40-50 years are champions of philosophical nothingness (Sartre and his woman etc.), the worst rationalism ever. But there are solid thinkers too, like Annales School for ex.

MoR: Believe me, insularity is a problem! It can be an asset in some ways, too, as I am sure the various English and Scottish successes help demonstrate. Close, but not too close, can have its advantages. Maybe half-awake Australia might one day demonstrate this too, especially with the shrinking of distance. You have no idea how much difference the internet makes. The fact I can tune into the deliberations of Lichanos, you MoR and Jahsonic, et al is a very fine thing!

Science is not enough. On leaving school, science trained, I craved a context. In my day science-culture was all dungeons and dragons or compulsory opera-superior (for want of knowing what else to do). There was something very hollow about it all. Mind you, my flirtation with political philosophy has left me (and maybe it is my fault) with a few interesting points to be counted on one hand, and genuine shock at the desperation of the rest.

The continental rationalist … has produced totalitarian minds.
Yes, this is true, and it fascinates me. Almost morbidly. I think Mark shares some of this feeling. (BTW, I am about to read a new selection of Robespierre’s writings in a book called Virtue and Terror!)

But there are solid thinkers too, like Annales School… Fernand Braudel rules!

A. WTH is blockhead-beef eaters?
I thought I explained this. “Beef Eater” is a sort of pejorative-affectionate term for the British, who love their roast beef. See the Gillray print I linked to, here: French Liberty – British Slavery

…such a poor thing, science as a religion… Yes, true. But of course, almost anything is pretty poor as a religion, except religion. Did I ever mention that when I was young I entertained a desire (tongue in cheek) to be pope? I liked the clothes and ritual, and the Latin. Started to teach myself to read it a bit. Failing that, I dreamed of being a canon lawyer, weaving crafty and learned arguments in favor of papal supremacy over the college of cardinals. Really, if I were to fall into religion, I think I’d be a Catholic. Such a good story, and all that art! Anyway, Stanislaw Lem once put it well: Science explains the universe to us; art, philosophy, religion, reconcile us to it.

As for Popper again, I’ll add I like his idea of open society, but I think his criticism of historicism to be … too logical … thence abstract.
Not sure what you’re onto here. His critique boils down to something pretty straightforward: He denies that historical investigation can reveal any overarching pattern, goal, or necessity to historical development. I agree with that completely.

BTW, Popper’s studies of historicism and society are rather separate from his epistemology of science, which has been criticized at times for being too abstract, i.e. too removed from how science is actually practiced. His student, Paul Feyeraband (Against Method) makes that point, but that book, though intriguing, seemed to me like an Oedipal rebellion against the powerful father-mentor. He makes some good points, but fails to score a victory in his zeal to rebel anarchically.

Lichanos: “Not sure what you’re onto here. His critique boils down to something pretty straightforward: He denies that historical investigation can reveal any overarching pattern, goal, or necessity to historical development. I agree with that completely.”

Popper was possibly referring to that Positivist historicism that Gramsci rejected because he considered it ‘encrusted’ – his words – with a naive faith in science. G sharply criticised the stupidity of those at the Comintern who saw the revolution as inevitable since to them history was ruled by necessity (which brought to being passive.)

“It is certain – he writes *here* – that prediction only means seeing the present and the past clearly as movement. Seeing them clearly: in other words, accurately identifying the fundamental and permanent elements of the process. But it is absurd to think of a purely “objective” prediction.”

[To understand the whole passage you need to read the 2 notes.]

It makes me think of beautiful Asimov’s Trilogy – I’m perhaps repeating myself – where A. imagines a super advanced historical science capable of predicting, even in a future 3000 years far from now, a peanut falling from a little girl’s hand in a Tel-Aviv kindergarten!

Popper was maybe closed in his library a bit and didn’t realise what was really happening in the concrete historicism field (or in the day-to-day scientist labs as you say). Gramsci instead came from the battle field of International politics in times of revolution. He received a Sardinian, South Italian & Piedmontese education, surely one of the biggest Italian brains of the last century. He lived 2 years in Russia, then came back and was first co-founder of the Communist Party of Italy and later became its leader, soon to be arrested by Mussolini.

In prison he wrote a treasure of Notebooks that go well beyond Marxism and are liberal (partially.) Together with Leopardi, he is our Montaigne. See Gramsci and neo-Gramscianism in the wiki. I am not a marxist but I owe a lot to him in my young years.

I read a bit of Gramsci long ago, (The Notebooks, I think) and I am aware of his freedom of thought as compared to doctrinaire communists. I don’t have much interest in reading Marxist critiques of Marxism, however – it all seems so obvious. To the writers, it’s of earth shaking importance because they are questioning The Faith. (Not that he isn’t worth reading, but a day has 24 hours only.)

Popper was maybe closed in his library a bit and didn’t realise what was really happening in the concrete historicism field…
You’ve lost me here… Popper wasn’t referring positivist thinkers – he was referring to Hegelians and Marxists.

Asimov imagines a super advanced historical science capable of predicting, even in a future 3000 years far from now, a peanut falling from a little girl’s hand…
A common theme in his writing. The basis of much of his Foundation trilogy, I believe. He wrote a short story about a society in which statistical science had reached such a high level of development that democratic elections were decided by polling the opinion of ONE selected citizen! This story was satirical, but often, he was not.

Talk about a naive faith in science!! The anecdote you quote is nothing more than the doctrine of predestination recycled, with human scientists as God. Our notions of causality and determinism are very mixed up, so these ideas are always popping up where they seem strange. I have a lengthy post that touches on many of these issues: Free Will and All That. I even break my taboo on discussing quantum effects.

But Marxists came from the Heghelian left, and many became positivist since Marx himself had a naive faith in science that is clearly of positivist origin.

Croce was first Marxist and he wrote a lot on that, on such crusts of faith … The Capital tries to demostrate the theory of plusvalue is a way ….
____________

Yes, this is true, and it fascinates me. Almost morbidly. I think Mark shares some of this feeling. (BTW, I am about to read a new selection of Robespierre’s writings in a book called Virtue and Terror!)

You might be interested in Gramsci’s reflection on Machiavelli then (though it is extremely rational); on the same page I linked to, Gramsci wrote:

“Another point which needs to be defined and developed is the “dual perspective” (see note 54 for that) in political action and in national life. The dual perspective can present itself on various levels, from the most elementary to the most complex; but these can all theoretically be reduced to two fundamental levels, corresponding to the dual nature of Machiavelli’s Centaur (note 55) — half-animal and half-human. They are the levels of force and of consent, authority and hegemony, violence and civilisation…”

Machiavelli in fact wrote in ‘Il principe':

“You should understand, therefore, that there are two ways of fighting: by law or by force. The first way is natural to men, and the second to beasts. But as the first way often proves inadequate one must needs have recourse to the second. So a prince must understand how to make a nice use of the beast and the man. The ancient writers taught princes about this by an allegory, when they described how Achilles and many other princes of the ancient world were sent to be brought up by Chiron, the centaur, so that he might train them his way. All the allegory means, in making the teacher half beast and half man, is that a prince must know how to act according to the nature of both, and that he cannot survive otherwise.”

Then note 55 quotes other related passages by Gramsci that are very stimulating. And he connects to your Jacobins too, to the Church and religion as instrumentum regni etc. *HE* ranges very wide, always, and in a supreme way. He is so darn mind expanding (and dialectic.)

…Marxists came from the Hegelian left, and many became positivist…
Well, positivism was a powerful current in 19th century thought, so I guess your statement is true. I doubt Marx or Lenin would have seen themselves as positivist, but maybe I’m wrong. Anyway, Lenin is a fine example of the naive faith in science you cite, a dogmatic and rather simpleminded notion of science. The icing on the cake was to insist that there was a Marxist-Leninist science, as well.

Gramsci is not a critique of Marxism. He is just Gramsci, something entirely new though coming from Marx. He summarizes all European and world history in a concrete way, althought Italy is a the centre of his reflection, but an Italy from Caesar to his day.His notes on pop culture are are fundamental btw. He provides new tools for a sociology of ideas. What is weird is that he is very studied in Japan and in India. The latter I can understand, since it was linked to Soviet Union for a while, but Japan, it is mysterious. Anglo-saxon neo-Gramscianism is instead a bit flat and too linked to Marxism to be interesting to me.
Gordon Brown said he was a pupil of Gramsci. If I recall well – I might be wring – he gave an impulse to have G translated.